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Transcript of UNCLASSIFIED 29 6 776'
UNCLASSIFIED29 1 7AD' 29 6 776'
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Western Management Science Institute
University of California * Los Angeles
UNIVERSITY OF CALIFORNIA, LOS ANGELES
GRADUATE SCHOOL OF BUSINESS ADMINISTRATION
WESTERN MANAGEWENT SCIENCE INSTITUTE
January 1963
Western Manegement Science InotituteWorking Paper No. 27
Management Sciences Research ProjectResea.ch Report No, 81
JOBSHUP-LKXE QUEUEING SYSTEMS
by
James R. Jackeon
Summary. The equillbrium joint probability distribution of queuelengths is obtained for a broad class of Jobehop-like "networks ofwaiting ines," where the mean arrival rate ot customers dependsalmost arbitrarily upon the number already present, and the meanservice rate at each service center depends almost arbitrarily uponthe queue length there. This extension of the author's earlier workis motivated by the obse:?vatizn that real production systems areusually subject to influences which make for increased stability bytending, as the amount of work-in-proceas grows, to reduce the rateat which work is injected or to increase the rate at whichprocessing takes place.
This paper combines and extends upon the content of Western
Management Science ILstitute Working Papers No. 10 and 19.
Acknowledgements. This work was supported partly by the Officeof Naval Research under Task 047-003t and partly by the WesternManagement Science Institute under a grant from the Ford Foundation.Reproduction In whole er in part is permitted for any purpose of theUnited States Government.
1,2JOBSHOP-LIKE QUEUEING SYSTEMS
by
James R. Jackson
Western Management Science InstituteGraduate School of Business Administration
UCLA
Summary. The equilibrium joint probability distribution of queue lengths isobtained for a broad class of jobshop-like "networks of waiting lines," wherethe mean arrival rate of customers depends almost arbitrarily upon the numberalready present, and the mean service rate at each service center dependsalmost arbitrarily upon the queue length there. This extension of theauthor's earlier work is motivated by the observation that real productionsystems are usually subject to influences which make for increasedstability by tending, as the amount of work-in-process grows, to reducethe rate at which new work is injected or to increase the rate at whichprocessing taks place.
1. Introduction
This paper provides the equilibrium joint probability distribution of
queue lengths for a bsoad class of queueing-theoretical models representing
multipurpose production systems composed of special-purpose service centers
(e.g, manufacturing jobshops). In the systems modelled, customers arrive
from time to time, each with a routing, which is an ordered list of service
centers where the customer must be served. An arriving customer joins the
queue at the first center on his routing, remains there until his service
there is completed, then goes to the second center on his routing (if any)
and remains there until served, and so forth, until all his required services
are completed, at which time he leaves the system.
1. This work was supported by the Office of Naval Research under Task047-003 and by the Western Management Science Institute Ford Foundationgrant. Reproduction in whole or in part is permitted for any purposeof the United States Government.
2. To be presented to the Joint International Meeting of the Institute ofManagement Sciences and the Operations Research Society of Japan, Tokyo,
August 21-24, 1963.
2
The customer arrivals are modelled as a generalized Poisson process,
whose parameter (mean arrival rate) varies almost arbitrarily with the
total number of customers already in the system. Service completions at
each center are also modelled as generalized Poisson processes, the parameter
(mean service rate) at each center varying almost arbitrarily with the queue
length there (number of customers in the queue, waiting and being served).
The generation of routings is modelled as a random walk among the names
of the service centers, which with probability one reaches a terminal state
marking the ends of routings. The mechanism for generating routings is
sufficiently general to include, for instance, cases where every conceivable
routing may occur, the possibility that all customers require services at
all centers in the same order, and instances in which each customer requires
service at just one of the centers.
The mathematical model does not explicitly reflect any particular rule
for deciding which, among several customers at a service center, is to be
served "next." The model and its analysis are, in fact, consistent with
all rules which do not depend (directly or indirectly) upon the future
routing or service time requirements of any customer. Perhaps the most
commonly used rule in this category, and the one whose operation is most
easily visualized, is the first-come, first-serve rule.
Results. The basic result of this paper is the equilibrium joint
probability distribution of the queue lengths at the several centers in the
jobshop-like queueing system just outlined. This result is stated as a
theorem in Section 4, following the detailed formulation of the basic model
and preliminary analysis in Sections 2 and 3. In Section 5, the theorem of
Section 4 is generalized to ;over cases where the immediate injection of a
3
new customer is triggered when the total number of customers falls below
a specified limit, or where a service is deleted if a queue grows beyond
a specified maximum length. Section 5 includes remarks about the theorem
and points to some of its implications.
Section 6 is devoted to special cases. Systems with constant customer-
arrival rate are considered; and it turns out that the main theorem of Section
5 reduces in this case to a statement that the equilibrium joint probability
distribution of queue lengths is identical with what would be obtained by
pretending that each individual service center is a separate queueing
system independent of the others. The constant arrival-rate case is further
specialized to the instance where each center is an "ordinary" multi-channel
3service system. Next, to clarify the effect of varying arrival rates,
attention is directed to a special case of the general model, in which all
Service rates are the same. Finally, systems are considered in which the
total number of customers present is held fixed.
Significance. The generalized customer-arrival process of the present
paper allows for the representation of systems whose "potential customers"
are generated by a stable Poisson process, but where the probability that a
potential customer will actually enter the system for service depends upon
how many customers are already there. In addition to the representation of
"ordinary" multiserver centers, the generalized service-completion process
also permits the treatment of systems at whose centers the number of servers
actually functioning depends almost arbitrarily upon the queue length there,
as well as plausible rough approximations to such phenomena as "speedups" and
"overtime work." The further generalizations (Section 5) of the customer-
3. Which was analyzed previously in the author's paper, "Networks of WaitingLines," Operations Research, vol. 5, no. 4 (Aug., 1957).
4arrival and service-completion processes make possible crude approximations
to such phenomena as the activation of "standby jobs" when a shopload falls
off, and "subcontracting" when a service center becomes overloaded.
2. Notations and Restrictions
The first business of this section is to specify and provide notations
for the parameters of the generic jobshop-like queueing system. As this
is done, these parameters are interpreted and restrictions are stated. The
model is mathematically formalized in Section 3.
States of the system. The number of service centers in the system will
be denoted by N , a positive integer; and the centers will be referred to
as Center 1, Center 2, etc. The states of the system are N-dimensional
vectors with non-negative integer components, the n-th component representing
the queue length at Center n. Such vectors will be called state vectors.
If k = (kl,k2, . * • , 2) is a state vector, then S(k) = k, + k2 + + k N
the total number of customers in the system when its state is k.
4Customer-arrival and service-completion processes. The customer-arrival
process is specified by a set of parameters,
L = (X(X) I K E [O,-)) .
The interpretation is that if the system Is in state k at time t , then
the probability that a customer will arrive between then and time t+h
is h \(S(k)) + o(h) . Similarly, the service-completion processes are
specified by a set of parameters,
M = (p(n,h) I n E [0, o) ;
with the interpretation that if k customers are present at Center nn
4. If a and b are integers with a < b,, then [a,w) denotes theset of integers not smaller than a ,-and [a,b] denotes the set ofintegers between a and b inclusive. The expresston o(h) denotesunspecified quantities such that as h tends to the limit zero, sodoes o(h)/h .
5
at time t , then the probability that the service of one of these
customers will be completed by time t+h is h p(n,k ) + o(h) . The
probability of two or more events (arrivals or completions) in a time
interval of length h is o(h) .
From their interpretations, the X(K) and the p(n,k) must be
non-negative, and also each p(n,O) = 0 . However, it is convenient, and
seems to result in no loss of interesting generality, to make somewhat
stronger assumptions, as follows:
ASSUMPTION (2.1). Either the I(K) are all positive; or for sono
non-negative K , X(K) > 0 if K < K but '(K)-= 0 if K > K ,
ASSUMPTION (2.2). The i(n,k) are all positive, except that
each p(n,O) = 0 .
(These assumptions may seem to exclude the possibly interesting case
where a service center is "shut down" if the queue length there falls to
some speoified lower limit. This case can, however, easily be biought into
this paper's analytical framework, by redefining the n-th component of the
state vector as the number of customers at Center n in excess of the lower
limit there, and appropriately translating the arguments of X(K) and
4(n,k) )
Generation of routings. The routing-generation process is specified
by a set of parameters,
R = (r(mpn) I m E[O,N] ; n E [1,11A.11)
The interpretation is that, for m and n Ell,N] : (i) the probability is
r(O,n) that Center n will be first on a routing; (ii) the probability is
r(m,N+l) that if Center m is i-th on a routing, then this i-th element is
the last one; (iii) the probability is r(m,n) that if Center m is i-th
on a routing, then there is an (i+l)-st element and it is Center n; and,
6
for completeness, (iv) the probability is r(O,N+l) that a routing will be
empty. It is easily seen that the probability of routing (nl, n2, . . ) , i)
is the product,
r(O,n I1) r(nl,n2, ) . . . r(nil,n i ) r(ni,,N+l)
Two assumptions are made:
ASSUMPTION (2.3). For each m E[O,N] , the set
( r(m,n) I n E[l,N+I]) is a probability distribution.
ASSUMPTION (2.4). The set of equations,
N(2.5) e(n) = r(O,n) + Z e(m) r(m,n) , n E [1,N] ,
-W 1
has a unique solution,.( e(n) I n E[l,N]) ; and the e(n) are all non-
negative.
Assumption (2.3) is necessary in view of the interpretation of the
r(m,n) . In the presence of (2.3), Assumption (2.4) can be shown to be
equivalent to the requirement that a routing terminate with probability one.
It is intuitively obvious (and can easily be proved) that e(n) is the
expected value of the number of appearances of Center n on a routing.
3. Mathematical Model
The informal interpretations of Section 2 are now given mathematical
substance by stating the conditional probabilities that the generic jobshop-
like queueing system will be in various states J = (Vlp.2 , iN)
at time t+h , given that it is in state k = (kl1 k2 . . . , k) at
time t . These probabilities are as follows:N N
1 - h \(S(k)) t r(O,n) - h E p(n,k ) (l-r(n,n)) + o(h) , if j = kn=l n=l n
h X(S(k)) r(Ox) + o(h) , if j = k except Jx = kX + 1
h p(x,k) r(x,N+l) + o(h) , if j=k except Jx =k - l;
h p(x,k ) r(x,y) + o(h) , if 3j=k except jx = k - IJ = k + ;
o(h - l ) , for 3 with IS(3) - S(i)I =.s > 1 ;
7
where x ii the second and third expressions runs through [1,N] ; and
x and y in the fourth expression run through [1,N] , omitting pairs
with x =y.
The first four of the above expressions reflect, respectively, the
ipossibility of no "events" occuring in the interval between times t and
t+h (except perhaps the completion of a service for a customer who then
requires another service at the same center), a customer arrival, the
completion of the last service on a customer's routing, and the completion
of one service for a customer who then requires another service at a different
center. The fifth expression reflects the need for at least IS(D) - S(0)
events to intervene between the appearances of states J and k .
The process defined by the procedin, oxprossions, with the notations
and subject to the assumptions of Section 2, will be referred to as
Jobshop-like queueing system (N,L,M,R), or more briefly as System (N,L,M,R).
Time-dependent state probabilities. Let P(k,t) be the probability
that System (NL,M,R) is in state k at time t ; and let P(k,t'Ij,t)
be the conditional probability of state R at time t' , given state J
at time t . From elementary probability theory,
P(k,t+h) = EP(j,t) P(i,t+hlj,t)
the sum extending over all state vectors J ; for each state vector k
If the conditional probabilities given previously are substituted in this
equation; and then P(k,t) is subtracted from both sides; and then both
sides are divided by h ; and thei,, finally, the limit is taken as h tends
to zero; the final result is the following differential-difference equation:
(3.1) aP(kt) = -[X(S(k)) + Zi-(n,k ) (1 - r (n,n))] P(k,t)dt n n
+ 0(S(k)-i) r(O,n) P(h(n),t)
+ !1(n,1; +1) r(n,+l) P(I(n),t)n n
+ =P(n,ik +1) r(n,m) P(3(m,n),t)ma n
8
where the. sums extend over (1,N] , except that pairs with m = n are
omitted from the double sum, and terms are omitted for which the vector
argument of P has a negative component; h(n) = k , except that its
n-th component is k - J. 1(n) = k , except that its n-th component
is kn + I ; and R(m,n) =k , except that its m-th component is km - 1
and its n-th component is k + 1.
4. Equilibrium for Jobshop-Like Queueing System (N,L,M,R)
By definition, an equilibrium state probability distribution for
System (N,L,M,R) is a probability distribution, (p(k)) , over state
Vectors k , such that P(k,t)np(k) specifies a (constant) solution
to equations (3.1). If such a distribution exists, it is unique, and
for each state vector k
lim P(k,t) = p(k)
the limits being independent of the "initial" state of the system.5 Thus,
the equilibrium state probability distribution provides a rather complete
description of the long-run-average behavior of the state of the system.
The theorem of this section gives a condition under which such a
distribution exists, and specifies the distribution.
5. That the P(k,t) approach limits follows from Theorem 2.9,p. 102, A.T. Barucha-Reid, Elements of the Theory of Markov
Processes and their Applications (blcGraw-Eill, 1960). Itcan be argued, using Assumptions (2.1) and (2.2) that everynon-transient state of System (N,L,M,R) communicates withevery other, and hence tinat if a limiting probabilitydistribution exists, it is unique. Th4 assertions thenfollow from the observation that the equilibrium stateprobability distribution is one possible "initial"distribution over state vectors.
9
The following notations will be used in stating the theorem, and
subsequently:K-1
(4.1) W(K) = T X(i) , for K = 0, 1, 2, .
i=ON k
(4.2) w(k) = TTn[e(n)/P(n,i)] , for k = (kl, k2, . . . , kN )n=l i=l
a state vector;
(4.3) T(K) = Ew(k) , summed over state vectors k with S(k) = K
gor K = 0, 1, 2, . . .
(4.4) Wl, if the sum converges,
= 0 , otherwise.
It is easy to verify that the sum in (4.4) converges to a positive number
or diverges to plus infinity.
The proof of the following theorem (after the preceding development)
is a straightforward matter of verifying that (4.6) does define a
probability distribution ("obvious"), and that equations (3.1) are
satisfied by this distribution (by direct substitution and a moderately
tedious, but routine, algebraic reduction):
THEOREM (4.5). If TT > 0, then a unique equilibrium state probability
distribution exists for Jobshop-like queueing system (N,L,M,R), and is
given by
(4.6) p(i) = TWOK) vl(S(0)
for k a state vector.
An example. The above theorem is extended in Section 5, and the
etatement of remarks and derivative conclusions will be deferred until this i.:
done. As a preliminary illustration of the theorem, consider a
6. Empty products are assigned the conventional value, +1 .
10
two-service-center jobshop-like queueing system for which the parameters
are as follows:7
(4.7) X(K) = a/(K + 1)x , where a > 0 and x 0
(4.8) P(1,k) = b k where b > 0 and y > 0
(4.9) P(2,k) = ck , where c > 0 and z 0
(4.10) r(0,1) = r(1,2) = r(2,3) = 1 ; the other r(m,n) = 0
The specification of R implies that each customer requires exactly two
services, the first at Center 1 and the second at Center 2; and it follows
that e(1) = e(2) = 1 .
After substituting in (4.1) through (4.4), it is not difficult to
verify that TT > 0 unless both (i) x = 0 and (ii) either y = 0 and
a/b > 1 or z = 0 and a/c - 1 . If rn> 0 , then Theorem (4.5) provides
the joint equilibrium probability distribution of queue lengths at the two
centers, as follows:
rr(a/b~ i (a/c) J
(4.11) p(i,J) - ,ja4 )x (ai/ )Y
The numerical value of rr can be obtained for specific values of the
parameters by using the fact that the p(i,j) must sum to one.
Variations upon this example will be used again in subsequent
sections.
5. Triggered Arrivals and Service Deletions
In this section, the concept of System (N,L,M,R) is generalized
to include cases where the immediate inje-:tion of a new customer is
7. The particular example used here was suggested by R. IV. Conwayand 17. L. Maxwell, "A Queueing Model with State Dependent ServiceRates," Journal of Industrial Engineering, vol. XII, no. 2 (1962);
which treats a queueing system equivalent to a single one of theexample's two interacting service centers.
11
triggered whenever the total number of customers falls below a specified
limit, or whore a service is deleted if a queue grows beyond a specified
maximum lenoh. The generalized model, to be referred to subsequently as
Jobihop-like queueing system (N,LM,R)*, or as System (NL,MR)*, is
identical with System (N,L,M,R), except for the following:
(1) There is some non-negative integer, K* ; such that the initial
state k satisfies S(k) > K* ; and if the last service on some customer's
routing is completed when the system is in a state k with S6) - K* I
then the arrival of a new customer occurs automatically at that instant.
(ii) For each Center n, there is a k n which may be a positive
integer or + o ;such that the initial state, k = (kl, I2, . * . , kX)satisfies kn < ka * for n E [1,N] ; and if a customer arrives at center n
when the queue length there is k n then the imainent service required
by that customer at Center n is deleted from his routing, and he proceeds
from there immediately, as if he had been served.0
It is implicit in the above statemei, ts that K* < k * + k2 * + . .4 .
If K* > 0 , then the X(K) for K < K* are obviously meaningless; ane.
if kn * < + ; , then the P(nk) for k - k n* are obviously meaningless.
System (N,L,M,R) is the special case of System (N,L,M,R)* with
K* = 0 and each k n* = -Fm . To formalize System (N,L,M,R)* would involve
rewriting the conditional probabilities (beginning of Section 3) for cases
where 8(k) = K* and/or one or more k n = * n and rewriting equations (3.1)
correspondingly. This will not be done here, because the modified equations
are of insufficient interest to justify the development of (necessarily
0. Or, more generally, some customer is emitted from the center;with the restriction that the choi.ce of the customer to beemitted must not depend upon the future routing or servicerequirements of any customer.
12
rather formidable) appropriate notations. But it is necessary to
give generalizations of equations (4.1) through (4.4) to prepare for
the generalization of Theorem (4.5):
(5.1) W*(K) = 0 , for K < K*
K-i
= n X(i) , for K > I*i=K*
N k(5.2) w*(i) = TY rn [e(n)/.(n,i)], if k < k * for n E [1,N]
n=1 i=1 n -1
= 0 , otherwise;
(5.3) T*(K) = Zw*(k) , summed over k with S(k) = K;
(5.4) 0* = ox*(K) T*(K)} - , if the sum converges,
= 0 , otherwise
If K* = 0 and each of the k n = +m , then equations (5.1) throughn
(5.4) specialize to equations (4.1) through (4.4), upon removal of
the asterisks.
The following theorem, which can be proved by substitution in the
modified version of equations (3.1), differs from Theorem (4.5) only in
the presence of the asterisks:
THEOREM (5.5). If Tr* > 0 , then a unique equilibrium state
probability distribution exists for Jobshop-like queueing system
(N,L,M,R)*, and is given by
(5.6) p(k) = 11* w*() l*(S(k),
for k a state vector.
Remarks. It is natural to conjecture that if r* = 0 , then no
equilibrium state probability distribution exists for System (N,LM,R)*,
but I have been unable to prove this except in special cases. If either
(M) there is some (finite) K such that ?(K) = 0 iorI X I{° , or
(ii) all of the k * are finite; then only finitely many terms in then
sum of (5.4) can be positive, so it is assured that rr* > 0, and the
theorem applies.
The "discovery" of Theorem (4.5) resulted from making a sequence
of "guesses's concerning more and more general jobshop-lihe queueing
systems, and proving successively more general versions of the theorem.
The way in which this took place may be suggested by the discussion of
certain special cases in Section 6.
Some corollary results. Suppose tnat System (N,L,M,R)* satisfies
the condition, Tr* > 0, so that Theorem (5.5) applies. Write p(S=K)
for the equilibrium probability that the total number of customers in the
system is K ; i.e., that the state x satisfies S(R) = K . If
p(S = K) > 0 , and k is a state vector, write p(kjK) for the
equilibrium conditional probability of state k , given that the total
number of customers in the system is K ; i.e., p(klIQ= p(k)/p(S=K)
if S(k) = K , but otherwise p (kiK) = 0 . It follows directly from the
theorem that:
(5.7) p(S = K) = r* T*(K) W*(K)
(5.8) p(kijS(k)) = w*(k)/T*(S(k)) , if p(S = S(k)) > 0
The probabilities (5.7) are of interest in themselves, because there
are instances where the total number of customers in the system is of
primary concern. They also lead to the overall mean arrival rate of
customers under equilibrium, E(X)
(5.9) E(X) =KO 7,(K) p(S = K)
= 1* ^ 1 lW*(K+l) T*(K)&MO0
14
Equation (5.9) is potentially of special interest, because it allows
the determination of "loss rates" when the variation in the (K) is
due to the failure of "potential customers" to enter the system for
service when too many customers are already present.
The main interest of the conditional probabilities (5.0) perhaps
stems from the observation that they do not depend upon the customer-arrival
process -- except for its role in determining whether 17* > 0 , and poswibly
in determining the values of C for which p(S = K) , 0 . In a sense, the
effect of particular customer-arrival-process parameters is concentrated upon
the determination of the p(S = K) .
Note also that the parameters of the routing-generation process
influence the equilibrium state probability distribution only by way of
the e(n) . It was remarked previously that e(n) is the expected number
of appearances of Center n on a routing.
Example. Consider a system identical with the example given at the
end of Section 4, except that for some K*, k1*, and k2* : Ci) if the
total number of customers in the system falls below K*, then a new
customer is instantly "created"; (ii) if a (k * + 1)-st customer joins the
queue at Center 1, he immediately moves on to Center 2, without being
served at Center 1; and (iii) if a (k * + 1)-st customer joins the queue
at Center 2, he immediately leaves the system without being served there.
Equation (4.11) is still valid for the p(i,j) such that i + j - K* ,
i < kI* , and j1< k2 ; but all other p(i,j) = 0 . There is a change,
of course, in the numerical value of IT (which might be replaced by TT*
for consistency with the notation of the present section).
15
6. Special Cases
The first part of this section is restricted to the special case
of System (N,L,M,R)* in which X(K) X(O) , to be referred to as the
constant-arrival-rate case of System (N,L,M,R)*. For this case, it is
convenient to define the following notations, for nE [1,N]
k(6.1) w(k) = l [X(O) e(n)/P(n,i)] , for k < k n
=0 , for k > kn
(6.2) pn(k) = wn(k)/f Wn(i) , if the sum converges,i=0
= 0 , otherwise,-for k = O, 1, 2,....
The following theorem is the direct specialization of Theorem (5.5) to
the constant-arrival-rate case:
THEOREM (6.3). If, in the constant-arrival-rate case of System
(N,LM,R)*, pn(0) > 0 for nE[i,N] ; then a unique equilibrium
state probability distribution exists for the system and is given by
the product,
(6.4) p(k1 , k 2 , . . . , kN ) = pl(kl) p2 (k 2 ) . . . PN(kN)
for (k,, k2, . . . , I- a state vector.
Interpretation. When Theorem (6.3) applies, it is immediate from
(6.2) that the p n(k) for fixed n form a probability distribution
over k = 0, 1, 2, . . . This distribution is, in fact, the equilibrium
distribution of queue length for the one-service-center queueing system
where customer arrivals form a Poisson process with mean rate N(0) e(n)
and whose service-completion process is identical to that of Center n
(which is well known, and also follows from the one-service-center case
of the theorem).
But e(n) is the mean number of appearances of Center n on a
routing in the generic system to which Theorem (6.3) refers, whence
X(O) e(n) is in fact the mean rate of customer arrivals at Center n
in this system. Thus, Theorem (6.3) can be interpreted as stating that
for the constant-arrival-rate case of System (NL,M,R)*, the equilibrium
probability distributions of queue lengths at the individual centers are
independent; and also each of these distributions is identical with that
for a one-service-center queuein3 system "similar" to the center concerned.
Further specializations. To represent an "ordinary" M(n)-channel
Center n, with exponentially distributed holding times whose mean is
1/1n , set P(n,k) = Pn min(k,M(n)) . If all centers are thus specialized
in Theorem (6.3), the result is the theorem of the paper cited in footnote 3.
In this instance, the sum of the w (i) can easily be expressed in closedn
form. Under this specialization, the one-service-center case of
Theorem (6.3) gives the well-known equilibrium distribution of queue lengtht
for an "ordinary" multi-channel service center with Poisson customer-
arrival process and exponentially distributed holding times.
Example. Suppose that in the example at the end of Section 4, x = 0
and, to assure the existence of equilibrium, assume it is false that
either y m 0 and a/b > 1 or z = 0 and a/c > I . Then
p(i,j) = pl(i) p2() ;
where pl(I) = p1 (0) (a/b) i/(i!)x and p2 () = P2 (O) (a/c) /(J:)y
the Pn(O) being determined by the conditions that Z , () 1
for n = 1, 2 .
Varying arrival rate, uniform service rates. To focus on the
effect of varying arrival rates, consider the special case of
System (N,L,M,R)* in which e(n)/P(n,i) = 1 for all i > 0 . Then
w*() = 1 for every state vector k . It is easy to sh-,4 that
17
T*(K) = N-l+K , for K = 0, 1, 2,....N - 1
It then follows that a sufficient (but not necessary) condition that
TT* > 0 is that lim sup X(K) < 1 . Suppose that TT* > 0 . If K
and j are two states, with S(k) = K > J = S(J) , where p(S = K)
and p(S = J) are positive; then, from Theorem (5.5):
K-ip(k)/p(j) = Li X(i) ;;
i=J
P(S = K) -1 (i + N) X(i)p(S =J) i i + 1
p(ijK) = (N -1"
The interested reader can improve his "feel" for the effect of varying
arrival rates by working out more specialized examples, including ones
in which p(S = X) > 0 for only finitely many different values of K
Total number of customers held fixed. Consider, finally, the special
case of System (N,L,M,R)* in which %(K) = 0 for K > K* . In this
system, customer arrivals occur when and only when other customers leave
the system; so the total number present will eventually remain fixed
at K*. It follows from Theorem (5.5) that p(k) = w*(k)/T*(S(k)) if
S(k) = K* , and otherwise p(k) = 0
In this instance, the model can also be interpreted as representing
a closed system through which the same customers circulate eternally.. In
this interpretation, the probability that a customer served at Center m
will next require service at Center n is r(m,n) + r(m,N+l) r(O,) ,
in the notation of the present paper. If the "givens" of such a system
include probabilities s(m,n) that a customer served at Center m will
next require service at Center n, for m and n Etl,N]; these
.. IC
probabilities can be appropriately converted for the application of
Theorem (5.5) by setting r(m,n) = s(m,n) for mE [I,N-1] and
n E [1,N] , r(NN+I) = 1 , and r(O,n) = s(N,n) for n E [1,N]
As a numerical example, consider a closed system made up of one
"machine operator" and two "repairmen," in which there are two
"machines." The machine operator functions like a service center of
the kind treated in the present paper; but the "service" he renders
is to cause the machine to "fail." The time to failure is exponentially
distributed with mean 10. When a machine fails it must be worked on
first by one repairman, then by the other, in a specified order. Each
repairman functions like a service center, with mean service rate 1/3
when he has one machine to work on, but with mean service rate 1/2 whea
he has two. Designating the machine operator as Center 1, and the
repairmen as Centers 2 and 3, what has been described is an instance
of System (N,L,M,R)*, with: N = 3 ; K* = 2 and %(k) = 0 for K> 2
P(1,1) = P(1,2) = 1/10 , P(2,1) = P(3,1) = 1/3 , and 1(2,2) = P(3,2) 1/2;
and r(0,l) = r(1,2) = r(2,3) = r(3,4) = 1 , whence e(l) = e(2) = e() = 1.
Using (5.2), (5.3), and the fact that p(k) = w*(k)/T*(2) for S(k) = 2
it turns out that the equilibrium state probabilities are as follows:
p(0,0,2) = p(0,2,0) = 6/131 ; p(0,l,l) = 9/181 ; p(l,0,l) = p(ll,O) =
30/131 ; p2,0,O) = 100/181 . It can be concluded, for instance, that
the long-run-average proportion of time during which the machine operator
will actually have a machine to operate is 16O/181 , or about 08.4 percent.
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